Density Functional Theory:
Deriving Phase Field Crystals
Simon Bignold
Supervisor: Christoph Ortner
University of Warwick
March 25, 2013
1
Wittkowski et al, arXiv:1207.0257, 2012
Simon Bignold Supervisor: Christoph Ortner
1
Outline
Model
Helmholtz Free Energy
One Particle Density
Introduction to DFT
Ideal Gas
Gradient Expansion
PFC
Further Work
Simon Bignold Supervisor: Christoph Ortner
Model
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Set-up
We have a fixed number of particles N with positions xi ∈ Λ ⊂ Rd .
For convenience we define
XN = (x1 , . . . , xN ) ∈ ΛN
and
β=
1
.
kB T
with Hamiltonian
X
HΛUN1 (XN ) =
U2 (xi − xj ) +
1≤i<j≤N
|
{z
}
inter-particle interaction U
N
X
U1 (xi )
|i=1 {z
}
external potential
U1 : Rd → Rd is the external potential U2 : Rd × Rd → R is the
interaction potential between particles.
Simon Bignold Supervisor: Christoph Ortner
Canonical Gibbs Ensemble
N
let ΓΛ = Λ × Rd
and equip it with the Borel σ -algebra on
β
ΓΛ .Then the probability measure γΛ,N
∈ P(Λ, BΛ ) with density
ρβΛ,N (XN ) =
exp[−βHΛUN1 (XN )]
N!ZΛ (β, N)
is called the canonical Gibbs ensemble.
Here ZΛ (β, N) is a normalisation factor known as the Partition
Function.
1
ZΛ (β, N) =
N!
Z
N
Y
exp[−βU1 (xi )]
ΛN i=1
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Y
1≤i<j≤N
exp[−βU2 (xi − xj )]dXN .
Free Energy
Simon Bignold Supervisor: Christoph Ortner
Helmholtz Free Energy
Free energy is minimised at equilibrium if temperature is held
constant.
We can also show 2
N
FβΛ [U1 ] = −β −1 ln[ZΛ (β, N)]

Z Y
N
1
= −β −1 ln 
exp[−βU1 (xi )]
N! ΛN
i=1
2
Simon Bignold Supervisor: Christoph Ortner

Y
1≤i<j≤N
exp[−βU2 (xi − xj )]dXN 
One-particle Density
Simon Bignold Supervisor: Christoph Ortner
One-particle Density
Three ways of doing this
• Integrating out N − 1 Variables
Z
Z
(1)
ρΛN (x) = N . . . ρβΛ,N (XN )dx2 . . . dxN
Λ
Λ
• Average over δ-functions
Z X
N
i
h
1
(1)
ρΛN (x) =
δ(x − xi ) exp −βHΛUN1 (XN ) dXN
N!ZΛ (β, N) ΛN
i=0
• Functional Derivative
N
(1)
ρΛN (x)
Z
Λ
=
(1)
δFβΛ [U1 ]
δU1 (x)
ρΛN (x)dx = N
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.
Introduction to Density Functional Theory
Simon Bignold Supervisor: Christoph Ortner
Introduction to DFT
We want to express free energy as the sum of a functional of
one-particle density only and another term.
(1)
U1 (x) is a conjugate variable to the one-particle density ρΛN (x).
Since free energy is a functional of the external potential, we can
use a Legendre transform to re-write the free energy.
Z
ΛN
Fβ [U1 ] = inf FHK [ρ̃(x)] + U1 (x)ρ̃(x)dx
ρ̃(x)
Λ
FHK is known as the Hohnberg-Kohn functional and the infimum is
over the space of absolutely continuous probability measure having
Lebesgue density.
Simon Bignold Supervisor: Christoph Ortner
Functional Taylor Expansion
We can split the Hohnberg-Kohn functional into two parts an ideal
gas part and an excess part
N
(1)
N
(1)
N
(1)
Λ
Λ
[ρΛN ]
[ρΛN ] + Fβ,exc
FβΛ [ρΛN ] = Fβ,id
We assume the existence of a reference density ρref and expand
the excess functional around this
(1)
N
Λ
FHK [ρΛN ] = Fβ,exc
[ρref ]
Z
Z
(1)
∞
ΛN
X
δ n Fβ,exc
[ρΛN ]
1
+
...
N
N
Λ (x ) . . . δρΛ (x ) n!
δρ
Λ
Λ
1
n
β
β
n=1
where
N
∆ρ(x) = ρΛβ (x) − ρref
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(x1 , . . . xn )
ρref
n
Y
i=1
∆ρ(xi )dxi
Ramakrishnan-Yussoff approximation
If the deviations from the reference density are small
∆ρ(x) 1
we can curtail the Taylor series. We chose to ignore terms higher
then i = 2 the i = 1 term vanishes as our potential is symmetric.
Thus we have
Z Z
(1)
ΛN
ΛN
−1
Fβ,exc [ρΛN ] = Fβ,exc [ρref ]−β
c (2) (x1 , x2 )∆ρ(x1 )∆ρ(x2 )dx1 dx2
Λ
Λ
where
c
(2)
ΛN [ρ(1) ] δ 2 Fβ,exc
ΛN
(x1 , x2 ) = −β ΛN
N
Λ
δρβ (x1 )δρβ (x2 ) ρref
3
3
Physics Review B, 19:5 p2775-2794,1979
Simon Bignold Supervisor: Christoph Ortner
(x1 , x2 )
Ideal Gas
Simon Bignold Supervisor: Christoph Ortner
Ideal Gas
In this case the internal potential, U2 , is zero
N

1 

ZΛ (β, N) =

N! 
Z


exp[−βV (x)]dx 

|Λ
{z
}
z(Λ)
Thus the free energy can be written as
N
FβΛ [U1 ] = β −1 (ln[N!] − N ln[z(Λ)]) .
Simon Bignold Supervisor: Christoph Ortner
Density Functional Form
We seek to re-write the free energy in a density functional form.
Using the functional derivative of the free energy we can find the
one-particle density
(1)
ρΛN (x) =
N exp[−βU1 (x)]
.
z(Λ)
Re-arranging we can find an expression for the external potential
" (1)
#
ρ
(x)z(Λ)
N
U1 (x) = −β −1 ln Λ
.
N
Simon Bignold Supervisor: Christoph Ortner
DFT Free energy
We therefore find
Z
Z
i
h
z(Λ)
(1)
(1)
(1)
−1
−1
.
ρΛN (x)U1 (x)dx = −β
ρΛN (x) ln ρΛN (x) dx−β N ln
N
Λ
Λ
We recall a generalisation of Stirling’s approximation 4
N
N
√
√
1
1
N
N
exp
≤ N! ≤ 2πN
exp
.
2πN
e
12N + 1
e
12N
Using this and that
energy as
N
FβΛ [U1 ]
R
(1)
Λ ρΛN (x)dx
= N we can re-write the free
Z
(1)
(1)
=β
ρΛN (x) ln[ρΛN (x)] − 1 dx
Λ
Z
(1)
+ ρΛN (x)U1 (x)dx + O(ln N).
−1
Λ
4
Robbins, The American Mathematical Monthly 62:1, pages 26-29,1955
Simon Bignold Supervisor: Christoph Ortner
The Small Deviations Regime
The part of the ideal gas functional not associated with the
external potential is
Z
(1)
(1)
(1)
−1
ΛN
Fβ,id [ρΛN ] = β
ρΛN (x) ln[ρΛN (x)] − 1 dx
Λ
If we know the reference density is constant and the deviation from
the density is small we can re-write the density as
N
ρΛβ (x) = ρref (1 + ψ(x))
Inserting this into our ideal gas equation we have
Z
(1)
ΛN
−1
Fβ,id [ρΛN ] = β
ρref (1 + ψ(x)) (ln[ρref (1 + ψ(x))] − 1) dx
Λ
Simon Bignold Supervisor: Christoph Ortner
Taylor Approximating the Ideal Gas
Using the Taylor expansion of the logarithm is
N
(1)
N
Λ
Λ
[ρref ]
[ρΛN ] = Fβ,id
Fβ,id
Z
ψ(x)2 ψ(x)3 ψ(x)4
a0 ψ(x) +
+ β −1 ρref
−
+
2
6
12
Λ
5
+ O ψ(x) dx
where
a0 = ln [ρref ]
Simon Bignold Supervisor: Christoph Ortner
Gradient Expansion
Simon Bignold Supervisor: Christoph Ortner
Small Deviation Regime
If we use the same approximation for the density in our expression
for the excess energy
Z Z
(1)
2
−1
ΛN
ΛN
c (2) (x1 , x2 )ψ(x1 )ψ(x2 )dx1 dx2
Fβ,exc [ρΛN ] = Fβ,exc [ρref ]−ρref β
Λ
Λ
where we have used that
∆ρ(xi ) = ρref ψ(x)
Using the definition of a convolution we have
Z (1)
(2)
ΛN
ΛN
2
−1
Fβ,exc [ρΛN ] = Fβ,exc [ρref ] − ρref β
c ∗ ψ (x1 )ψ(x1 )dx1
Λ
Simon Bignold Supervisor: Christoph Ortner
Fourier Expansion
Using that the fourier transform of a convolution is the product of
the fourier transforms of the functions in the convolution
Z
h
i
(1)
ΛN
2
−1
ΛN
F−1 ĉ (2) (k)ψ̂(k) ψ(x1 )dx1
Fβ,exc [ρΛN ] = Fβ,exc [ρref ] − ρref β
Λ
c (2)
we expand
as a Taylor series around k = 0 and use that odd
terms vanish by symmetry of c (2)
" ∞
#
Z
X
N
N
Λ
Λ
Fβ,exc
[U1 ] = Fβ,exc
[ρref ]−ρ2ref β −1 F−1
c2m k 2m ψ̂(k) ψ(x1 )dx1
Λ
m=0
Using that
h
i
F−1 k 2m ψ̂(k) = (−1)m ∇2m ψ(x)
we have
(1)
ΛN
Fβ,exc
[ρΛN ]
=
ΛN
Fβ,exc
[ρref ] − ρ2ref β −1
Simon Bignold Supervisor: Christoph Ortner
Z
ψ(x1 )
Λ
∞
X
m=0
c2m ∇2m ψ(x1 )dx1
PFC
Simon Bignold Supervisor: Christoph Ortner
Approximate Functional
We can re-combine our ideal gas functional and our excess energy
functional to give
Z
∞
X
(1)
FHK [ρΛN ] = FHK [ρref ] − ρ2ref β −1 ψ(x1 )
c2m ∇2m ψ(x1 )dx1
Λ
m=0
ψ(x)2 ψ(x)3 ψ(x)4
−
+
a0 ψ(x) +
+ β −1 ρref
2
6
12
Λ
5
+ O ψ(x) dx
Z
following 5 we curtail at fourth order in both ψ and the gradient.
The functional minus the part evaluated at the reference density is
Z
2
X
(1)
2
−1
∆FHK [ρΛN ] ≈ −ρref β
ψ(x1 )
c2m ∇2m ψ(x1 )dx1
Λ
+ β −1 ρref
a0 ψ(x) +
Λ
5
m=0
Z
ψ(x)2 ψ(x)3 ψ(x)4
−
+
dx
2
6
12
Stefanovic, Grant and Elder, Physics Review B, 75, 064107, 2007
Simon Bignold Supervisor: Christoph Ortner
PFC Functional and Equation
discarding the linear terms we have
Z
ψ(x)2
(1)
−1
A
+ Bψ(x)∇2 ψ(x) + C ψ(x)∇4 ψ(x)
∆FHK [ρΛN ] ≈ β ρref
2
Λ
ψ(x)3 ψ(x)4
−
+
dx
6
12
The classical PFC functional is given by an appropriate choice of
constants and absorbing the cubic term as
Z
2
ψ̃
ψ̃ ψ̃ 4
FPFC [ψ̃] =
∇2 + 1 ψ̃ − δ +
dx̃
2
4
Ω 2
The PFC Equation is given
6
by H −1 gradient flow
δFPFC [ψ̃]
δ ψ̃(x)
2
= ∇2 ∇2 + 1 ψ̃ − δ ψ̃ + ψ̃ 3
ψt = ∇2
6
Wirth and Elsey, Pre-print, 2012
Simon Bignold Supervisor: Christoph Ortner
Results
√
for the choice δ = 0.9 and ψ = 0.5 on a 6π × 4 3π domain with
periodic boundary conditions we obtain a hexagonal lattice
by altering the value of ψ we can obtain a constant density domain
(ψ > 0.5) or a striped pattern (ψ < 0.5).
Simon Bignold Supervisor: Christoph Ortner
Further Work
My Website
Defects
Surface Energy
Optimisation
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Acknowledgements
I would like to thank Christoph Ortner and Stefan Adams for all
their help and advice.
Simon Bignold Supervisor: Christoph Ortner